Embedding into bipartite graphs

The conjecture of Bollob\'as and Koml\'os, recently proved by B\"ottcher, Schacht, and Taraz [Math. Ann. 343(1), 175--205, 2009], implies that for any $\gamma>0$, every balanced bipartite graph on $2n$ vertices with bounded degree and sublinear bandwidth appears as a subgraph of any $2n$-vertex graph $G$ with minimum degree $(1+\gamma)n$, provided that $n$ is sufficiently large. We show that this threshold can be cut in half to an essentially best-possible minimum degree of $(\frac12+\gamma)n$ when we have the additional structural information of the host graph $G$ being balanced bipartite. This complements results of Zhao [to appear in SIAM J. Discrete Math.], as well as Hladk\'y and Schacht [to appear in SIAM J. Discrete Math.], who determined a corresponding minimum degree threshold for $K_{r,s}$-factors, with $r$ and $s$ fixed. Moreover, it implies that the set of Hamilton cycles of $G$ is a generating system for its cycle space.Comment: 16 pages, 2 figure

Trees and graph packing

In this thesis we investigate two main topics, namely, suffix trees and graph packing problems. In Chapter 2, we present the suffix trees. The main result of this chapter is a lower bound on the size of simple suffix trees. In the rest of the thesis we deal with packing problems. In Chapter 3 we give almost tight conditions on a bipartite packing problem. In Chapter 4 we consider an embedding problem regarding degree sequences. In Chapter 5 we show the existence of bounded degree bipartite graphs with a small separator and large bandwidth and we prove that under certain conditions these graphs can be embedded into graphs with minimum degree slightly over n/2

Survey of Graph Embeddings into Compact Surfaces

A prominent question of topological graph theory is what type of surface can a nonplanar graph be embedded into? This thesis has two main goals. First to provide a necessary background in topology and graph theory to understand the development of an embedding algorithm. The main purpose is developing and proving a direct constructive embedding algorithm that takes as input the graph with a particular order of edges about each vertex. The embedding algorithm will not only determine which compact surface the graph can be embedded into, but also determines the particular embedding of the graph on the surface. The embedding algorithm is then used to investigate surfaces into which trees and a class of the complete bipartite graphs can be embedded. Further, the embedding algorithm is used to investigate non-surface separating graph embeddings